There are many deep results (long-standing, famous, etc.) for which only one approach is known. Can you give examples of deep results, for which several genuinely different proofs have been found? With the example of the prime number theorem in mind, it would be interesting to know if these proofs are all considered equally hard, or some are significantly easier or, perhaps, even surprisingly easy.

Edit: I will narrow the definition of "depth" to the results which (a) occupy an important, if not central, place in a field, (b) are clearly attributable to an original paper (please cite).

Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How hard is the proof from the perspective of the non-expert in the field? In the opposite direction, can you give examples of important results, for which several genuinely different proofs were found? Are these proofs all considered equally hard, or some are much easier or, perhaps, even surprisingly easy?

For example, Carleson's theorem has more than one proof and, although the latest proof is much simplified, it is probably still quite technical. Poincare conjecture has one known proof, not easily accessible to non-experts. Kadison-Singer problem has one known proof, accessible to non-experts with a little effort. Capset problem has one known proof, fully accessible to non-experts. The last three examples are relatively recent results, and some of them are likely to remain the only known approaches for a long time, while maybe not others. Instead of trying to guess the future, what are interesting examples that are less recent?

There are many deep results (long-standing, famous, etc.) for which only one approach is known. Can you give examples of deep results, for which several genuinely different proofs have been found? With the example of the prime number theorem in mind, it would be interesting to know if these proofs are all considered equally hard, or some are significantly easier or, perhaps, even surprisingly easy.

There are many deep results (long-standing, famous, etc.) for which only one approach is known. Can you give examples of deep results, for which several genuinely different proofs have been found? With the example of the prime number theorem in mind, it would be interesting to know if these proofs are all considered equally hard, or some are significantly easier or, perhaps, even surprisingly easy.

Edit: I will narrow the definition of "depth" to the results which (a) occupy an important, if not central, place in a field, (b) are clearly attributable to an original paper (please cite).